General question here, I understand the purpose of using the harmonic mean to generate the f1 score for model evaluation. I'm not exactly sure though why we don't just take the product of precision and recall (precision x recall) and just use that number instead? What is the benefit of a harmonic mean over just the product of the rates?
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Recall the definitions of precision and recall: $$p=\frac{TP}{TP+FP}, r=\frac{TP}{TP+FN}$$
Both of them have meaning and focus on false positives or negatives. On the other hand, $F_1$ score seeks a common ground between them and has a meaning as well:
$$F_1=\frac{TP}{TP+\frac{1}{2}(FP+FN)}$$ where instead of taking only the false positives or negatives, their average is taken as false classifications.
All of the three formulas, in some way, define their true and falses and calculate success.
Of course, one could use $pr$ (or another measure) as a means for model selection, but note that you wouldn't reach a similar meaning with either $(p+r)/2$ or $pr$.
